Problem: Solve for $x$ : $ 8|x + 10| - 2 = 3|x + 10| + 4 $
Solution: Subtract $ {3|x + 10|} $ from both sides: $ \begin{eqnarray} 8|x + 10| - 2 &=& 3|x + 10| + 4 \\ \\ { - 3|x + 10|} && { - 3|x + 10|} \\ \\ 5|x + 10| - 2 &=& 4 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 5|x + 10| - 2 &=& 4 \\ \\ { + 2} &=& { + 2} \\ \\ 5|x + 10| &=& 6 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x + 10|} {{5}} = \dfrac{6} {{5}} $ Simplify: $ |x + 10| = \dfrac{6}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 10 = -\dfrac{6}{5} $ or $ x + 10 = \dfrac{6}{5} $ Solve for the solution where $x + 10$ is negative: $ x + 10 = -\dfrac{6}{5} $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& -\dfrac{6}{5} \\ \\ {- 10} && {- 10} \\ \\ x &=& -\dfrac{6}{5} - 10 \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{6}{5} {- \dfrac{50}{5}} $ $ x = -\dfrac{56}{5} $ Then calculate the solution where $x + 10$ is positive: $ x + 10 = \dfrac{6}{5} $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& \dfrac{6}{5} \\ \\ {- 10} && {- 10} \\ \\ x &=& \dfrac{6}{5} - 10 \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{6}{5} {- \dfrac{50}{5}} $ $ x = -\dfrac{44}{5} $ Thus, the correct answer is $x = -\dfrac{56}{5} $ or $x = -\dfrac{44}{5} $.